7 Part I CDMA Power Control Can maximize N i=1 SIR i with centralized control. (HCM, 2004) Since centralized control is not feasible for complex systems, how can such systems be optimized using decentralized control? Idea: Use large population properties of the system together with basic notions of game theory. Caines, 2009 p.7

9 Part II Statistical Mechanics A foundation for thermodynamics was provided by the Statistical Mechanics of Boltzmann, Maxwell and Gibbs. Basic Ideal Gas Model describes the interaction of a huge number of essentially identical particles. SM explains very complex individual behaviours by PDEs for a continuum limit of the mass of particles Animation of Particles Caines, 2009 p.9

15 Part II Statistical Mechanics Control of Natural Entropy Increase Feedback Control Law (Non-physical Interactions): At each collision, total energy of each pair of particles is shared equally while physical trajectories are retained. Energy is conserved Animation of Particles Caines, 2009 p.15

16 Part II Key Intuition A sufficiently large mass of individuals may be treated as a continuum. Local control of particle (or agent) behaviour can result in (partial) control of the continuum. Caines, 2009 p.16

17 Part III Game Theoretic Control Systems Game Theoretic Control Systems of interest will have many competing agents A large ensemble of players seeking their individual interest Fundamental issue: The relation between the actions of each individual agent and the resulting mass behavior Caines, 2009 p.17

24 Part III An Example: School of Sardines School of Sardines Caines, 2009 p.24

25 Part III Names Why Mean Field Theory? Standard Term in Physics for the Replacement of Complex Hamiltonian H by an Average H Notion Used in Many Fields for Approximating Mass Effects in Complex Systems Shall use Mean Field as General Term and Nash Certainty Equivalence (NCE) in Specific Control Theory Contexts Caines, 2009 p.25

26 Part III Names Why Nash Certainty Equivalence Control (NCE Control)? Because the equilibria established are Nash Equilibria and The feedback control laws depend upon an "equivalence" assumed between unobserved system properties and agent-computed approximations e.g. θ 0 and ˆθ N (which are exact in a limit). Caines, 2009 p.26

29 Part III Key Features of NCE Dynamics Under certain conditions, the mass effect concentrates into a deterministic - hence predictable - quantity m(t). A given agent only reacts to its own state and the mass behaviour m(t), any other individual agent becomes invisible. The individual behaviours collectively reproduce that mass behaviour. Caines, 2009 p.29

38 Part III NCE Control: Key Observations The information set for NCE Control is minimal and completely local since Agent A i s control depends on: (i) Agent A i s own state: x i (t) (ii) Statistical information F(θ) on the dynamical parameters of the mass of agents. Hence NCE Control is truly decentralized. All trajectories are statistically independent for all finite population sizes N. Caines, 2009 p.38

39 Part III Key Intuition: Fixed point theorem shows all agents competitive control actions (based on local states and assumed mass behaviour) reproduce that mass behaviour Hence: NCE Stochastic Control results in a stochastic dynamic Nash Equilibrium Caines, 2009 p.39

63 Future Directions Further development of Minyi Huang s large and small players extension of NCE Theory Egoists and altruists version of NCE Theory Mean Field stochastic control of non-linear (McKean-Vlasov) systems Extension of SAC MF Theory in richer game theory contexts Development of MF Theory towards economic and biological applications Development of large scale cybernetics: Systems and control theory for competitive and cooperative systems Caines, 2009 p.63

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